不规则区域上含时变系数Stokes流的一种简单有限差分方法
A simple finite difference method for time-dependent, variable coefficient Stokes flow on irregular domains
AI总结：
提出一种基于变分形式的嵌入式边界有限差分方法，用于模拟不规则自由表面和移动固体边界上的含时Stokes流，仅需每时间步求解一个稀疏对称正定线性系统。
摘要：
我们提出了一种简单高效的变分有限差分方法，用于模拟存在不规则自由表面和移动固体边界的含时Stokes流。该方法在交错笛卡尔网格上采用嵌入式边界方法，避免了昂贵的网格重划分操作，可应用于二维和三维流动。它使用全隐式后向欧拉积分以保证稳定性，支持空间变化的密度和粘度，每时间步仅需求解一个稀疏对称正定线性系统。通过将问题表达为变分形式，利用自然边界条件隐式支持不规则区域。实际中，离散化仅需中心有限差分模板和每个单元的体积分数，易于实现。变分形式还允许推广到其他力学耦合，同时保持稀疏对称正定矩阵。我们在二维的一系列解析测试案例上展示了速度在L1和L∞范数下的一致一阶收敛性。此外，我们将该方法作为简单Navier-Stokes求解器的一部分，证明其能够再现中等粘度牛顿液体在二维和三维中的特征射流屈曲现象。
英文摘要：
We present a simple and efficient variational finite difference method for simulating time-dependent Stokes flow in the presence of irregular free surfaces and moving solid boundaries. The method uses an embedded boundary approach on staggered Cartesian grids, avoiding the need for expensive remeshing operations, and can be applied to flows in both two and three dimensions. It uses fully implicit backwards Euler integration to provide stability and supports spatially varying density and viscosity, while requiring the solution of just a single sparse, symmetric positive-definite linear system per time step. By expressing the problem in a variational form, challenging irregular domains are supported implicitly through the use of natural boundary conditions. In practice, the discretization requires only centred finite difference stencils and per-cell volume fractions, and is straightforward to implement. The variational form further permits generalizations to coupling other mechanics, all the while reducing to a sparse symmetric positive definite matrix. We demonstrate consistent first order convergence of velocity in L1 and Linf norms on a range of analytical test cases in two dimensions. Furthermore, we apply our method as part of a simple Navier-Stokes solver to illustrate that it can reproduce the characteristic jet buckling phenomenon of Newtonian liquids at moderate viscosities, in both two and three dimensions.
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